Question

Implement the following logic function using a 2-to-1 multiplexer via Shannon's Expansion (expansion over one variable). You need to show your immediate steps.

f = x₁⊕x₂ + x₁x₃

Answer 1

To implement the given function f = x₁⊕x₂ + x₁x₃ using a 2-to-1 multiplexer, we expand f by Shannon's theorem over x₁. For x₁=0, we get f = x₂, and for x₁=1, f = ˉx₂ + x₃. Set the multiplexer inputs accordingly and use x₁ as the select line.

Step-by-step explanation:

The question pertains to the implementation of a logic function using a 2-to-1 multiplexer with the help of Shannon's Expansion theorem. The function to be implemented is: f = x₁⊕x₂ + x₁x₃.

First, let us expand the function over variable x₁.
When x₁ is 0, the function reduces to f = x₂ because x₁⊕x₂ becomes x₂ and x₁x₃ becomes 0.
When x₁ is 1, the function reduces to f = ˉx₂ + x₃, because x₁⊕x₂ becomes ˉx₂ and x₁x₃ stays x₃.

Therefore, a 2-to-1 multiplexer can implement f with its data inputs set as follows:

• Input 0 set to x₂ (since when x₁=0, f = x₂)
• Input 1 set to ˉx₂ + x₃ (since when x₁=1, f = ˉx₂ + x₃)

The select line of the multiplexer should be connected to x₁, thus selecting between the two input equations based on the value of x₁.